SRC/zlasyf_rk.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/zlasyf_rk.f

okay bestie, gather round, because this one is a whole LAPACK icon and she has ✨layers✨. The manual name is SRC/zlasyf_rk.f, and yes she's a Fortran source file page — Doxygen-generated, fancy.

✨ SYNOPSIS ✨

🫶 Functions/Subroutines

subroutine zlasyf_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)

ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. That's the whole vibe. 🎀

🔥 Function/Subroutine Documentation 🔥

subroutine zlasyf_rk (character uplo, integer n, integer nb, integer kb,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( *
    ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w,
    integer ldw, integer info)

ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. she's giving "auxiliary routine with main character energy" ngl 💅

🌸 Purpose 🌸

ZLASYF_RK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form: A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ( 0 U22 ) ( 0 D ) ( U12T U22T ) A = ( L11 0 ) ( D 0 ) ( L11T L21T ) if UPLO = 'L', ( L21 I ) ( 0 A22 ) ( 0 I ) where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

so she's literally the backstage icon that ZSYTRF_RK calls when it needs the heavy lifting done. blocked code, Level 3 BLAS, the works. 💖

🦋 Parameters 🦋

  • UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

pick your lane bestie, 'U' or 'L', those are the only two options, no third thing. 🙈

  • N
N is INTEGER The order of the matrix A. N >= 0.

how big is she? N. and N >= 0 — zero is allowed, don't be weird about it.

  • NB
NB is INTEGER The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.

NB should be at least 2 — the manpage says should, but like, take the hint. 2-by-2 pivot blocks need room to breathe. 🫶

  • KB
KB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

she's the actual answer, not the wishlist. 💫

  • A
A is COMPLEX*16 array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, contains: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

the main character herself. note that on exit she is not the same matrix you handed in — she's been through a glow-up. 🙈✨

  • LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

LDA >= max(1,N). respect the leading dimension or she will not respect you. 💀

  • E
E is COMPLEX*16 array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.

lowkey the unsung hero of this whole routine, she holds the off-diagonal bits so A doesn't have to. and that NOTE? read it. E(k) gets set to 0 for 1-by-1 blocks, both UPLO cases, no exceptions, periodt. 🔥

  • IPIV
IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If IPIV(k) = k, no interchange occurred. b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs). 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k. d) NOTE: Any entry IPIV(k) is always NONZERO on output.

okay this is the big one, the receipts section. lemme break the vibe down for you:

  • magnitude of IPIV(k) = which row/column got swapped with the k-th one 🌈
  • UPLO decides the order the interchanges happened in
  • the sign of IPIV tells you the block structure of D (1-by-1 vs 2-by-2 blocks)
  • UPLO = 'U': factorization order has k going down from N to 1
  • UPLO = 'L': factorization order has k going up from 1 to N
  • positive entry = 1-by-1 block; a pair of consecutive negatives = 2-by-2 block
  • negative entries appear ONLY in pairs, that's a rule, no cap 🫶
  • UPLO = 'U': ABS( IPIV(k) ) <= k; UPLO = 'L': ABS( IPIV(k) ) >= k — yes they're different, read carefully bestie 😭
  • any entry IPIV(k) is always NONZERO on output — she does not do zeros. periodt. 💅
  • W
W is COMPLEX*16 array, dimension (LDW,NB)

that's it. that's the description. workspace icon, no notes. ✨

  • LDW
LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).

LDW >= max(1,N) — same energy as LDA, don't test her. 💀

  • INFO
INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.
⚠️ bestie please read this part: if INFO > 0 the factorization still completed, but D is exactly singular and you will get division by zero if you try to use it to solve a system of equations. that's not a vibe check, that's a real warning. 🚨
and the NOTE at the end: INFO only stores the first singularity. any subsequent ones don't get recorded, even though the factorization always completes. so INFO = 1 might not be the whole story. 🙈

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

the whole squad really said "we built this" 🫶

🧚 Contributors: 🧚

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

icons, all of them, giving credit where credit is due 💜

Definition at line 260 of file zlasyf_rk.f.

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

(yeah, Doxygen wrote this one, she's a bot but she ate 💅)

and that's the whole page bestie!! LAPACK, Version 3.12.0, SRC/zlasyf_rk.f(3) — go forth and pivot responsibly 🎀✨🦋

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